Kolyvagin systems and Iwasawa theory of generalized Heegner cycles

Abstract

Iwasawa theory of Heegner points on abelian varieties of ${GL}_2$ type has been studied by, among others, Mazur, Perrin-Riou, Bertolini, and Howard. The purpose of this article is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne’s $p$-adic representation attached to a modular form of even weight greater than $2$. In this setting, the role of Heegner points is played by higher-dimensional Heegner-type cycles that have been recently defined by Bertolini, Darmon, and Prasanna. Our results should be compared with those obtained, via deformation-theoretic techniques, by Fouquet in the context of Hida families of modular forms.